Basically broadening has to do with randomness and disorder. Don't look at mean free paths.
It affects spectroscopy in huge ways; for instance an excessively broad line will have so low intensity that it will not be observable(this is one of the components of continuum lowering).
It also affects reabsorption, it affects the coherence properties of lasers and many others.

Plasma polarization shift is a controversial notion, based on a static picture. It is about the extra charge inside the emitter wavefunction that shifts the levels and hence the transition energy. The problem is that that extra charge inside the emitter wavefunction is not a static charge, but plasma electrons flying by. So if one accounts for electron dynamics correctly, it just is not there.